Lesson 4

Scaling and Area

Problem 1

A rectangle with area 12 square units is dilated by a scale factor of $$k$$. Find the area of the image for each given value of $$k$$.

1. $$k=2$$
2. $$k=5$$
3. $$k=1$$
4. $$k=\frac14$$
5. $$k=1.2$$

Solution

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Problem 2

The area of a circle of radius 1 is $$\pi$$ units squared. Use scaling to explain why the area of a circle of radius $$r$$ is $$\pi r^2$$ units squared.

Solution

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Problem 3

Trapezoid $$A’B’C’D$$ was created by dilating trapezoid $$ABCD$$ using $$D$$ as the center of dilation.

1. What was the scale factor of the dilation?
2. Based on the scale factor, how many copies of $$ABCD$$, including the original, should fit inside $$A’B’C’D$$?
3. How can you see your answer to these questions in the diagram?

Solution

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Problem 4

Each image shows a quadrilateral in a plane. The quadrilateral has been dilated using a center above the plane and a scale factor between 0 and 1. Estimate the scale factor that was used for each dilation.

Solution

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(From Unit 5, Lesson 3.)

Problem 5

Select the solid whose cross sections are dilations of some two-dimensional shape using a point directly above the shape as a center and scale factors ranging from 0 to 1.

A:

cone

B:

cube

C:

cylinder

D:

triangular prism

Solution

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(From Unit 5, Lesson 3.)

Problem 6

Select all figures for which at least one cross section is a circle.

A:

triangular pyramid

B:

square pyramid

C:

rectangular prism

D:

cube

E:

cone

F:

cylinder

G:

sphere

Solution

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(From Unit 5, Lesson 2.)

Problem 7

If the two-dimensional figures are rotated around the vertical axes of rotation shown, what solids are formed?

Solution

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(From Unit 5, Lesson 1.)

Problem 8

Tyler and Jada wish to find the value of $$x$$, the length of side $$BC$$ in this triangle. Tyler decides to set up the equation $$\tan(56)=\frac8x$$. Jada says she prefers an equation that has $$x$$ in the numerator. What is an equation she could use instead?

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Solution

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(From Unit 4, Lesson 8.)

Problem 9

Triangles $$ACD$$ and $$BCD$$ are isosceles. Angle $$DBC$$ has a measure of 110 degrees and angle $$BDA$$ has a measure of 22 degrees. Find the measure of angle $$BAC$$.

Solution

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(From Unit 2, Lesson 6.)